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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Assume the solution in the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A e^{2 x}.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
4 A e^{2 x}-2 A e^{2 x}-2 A e^{2 x}=e^{2 x} \to (4 A-2 A-2 A) e^{2 x}=e^{2 x}
\end{equation*}
</div>
<p class="continuation">The equation can never be satisfied which means the assumed form is wrong! The reason is that <span class="process-math">\(e^{2 x}\)</span> is a solution of the corresponding homogeneous equation. Assume that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A x e^{2 x}.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;4 A e^{2 x}+4 A x e^{2 x}-A(e^{2 x}+ 2 x e^{2 x})-2 A x e^{2 x}=e^{2 x}\\
&amp;\to (4A-2A-2A) x e^{2 x}+(4 A-A) e^{2 x}=e^{2 x}\\
&amp;\to 3 A=1 \to A=\frac{1}{3}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=\frac{1}{3} x e^{2 x}.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_6.html#p-130" class="internal">in-context</a></span>
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